where \(F\)  is in units of \(\lambda \equiv \frac{N_b r_0}{\gamma }\) , \(r_0\)  is
the classical particle radius, \(\Gamma _s(P) \equiv \Gamma (s,P)\) 
denotes the upper incomplete gamma function [1]}, and
\(\overline{\gamma }\)  = 0.577216 is Euler's constant. The
corresponding beam–beam kick is as follows:
\(& \Delta x^{\prime } \equiv \frac{\mathrm {d}}{\mathrm {d}x} F(x) = \frac{\partial F}{\partial P} \frac{\mathrm {d}P}{\mathrm {d}x} = \numero \\=& \frac{ 2(x+n_x\sigma ) }{ (x+n_x\sigma )^2+ (n_y\sigma )^2 }\left[ 1-\mathrm {e}^{- \frac{ (x+n_x\sigma )^2+ (n_y\sigma )^2 }{2\sigma ^2} } \right] .\) 
